Alternating Automata on Data Trees and XPath Satisfiability

📝 Abstract

A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (“datum”) from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language.

💡 Analysis

A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (“datum”) from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language.

📄 Content

arXiv:0805.0330v4 [cs.LO] 14 Jun 2010 Alternating Automata on Data Trees and XPath Satisfiability MARCIN JURDZI´NSKI and RANKO LAZI´C Department of Computer Science, University of Warwick, UK A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (“datum”) from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Formal Languages—Decision problems; F.1.1 [Computation by Abstract Devices]: Models of Computation—Automata; H.2.3 [Database Management]: Languages—Query languages General Terms: Algorithms, Verification 1. INTRODUCTION Context. Logics and automata for words and trees over finite alphabets are rel- atively well-understood. Motivated partly by the search for automated reasoning techniques for XML and the need for formal verification and synthesis of infinite- state systems, there is an active and broad research programme on logics and au- tomata for words and trees which have richer structure. Initial progress made on reasoning about data words and data trees is summarised in the survey by Segoufin [2006]. A data word is a word over Σ × D, where Σ is a finite alphabet, and D is an infinite set (“domain”) whose elements (“data”) can only be compared for equality. Similarly, a data tree is a tree (countable, unranked and ordered) whose every node is labelled by a pair in Σ × D. First-order logic for data words was considered by Boja´nczyk et al. [2006], and related automata were studied further by Bj¨orklund and Schwentick [2007]. The logic has variables which range over word positions ({0, . . . , l −1} or N), a unary predicate for each letter from the finite alphabet, and a binary predicate x ∼y which denotes equality of data labels. FO2(+1, <, ∼) denotes such a logic with two variables and binary predicates x + 1 = y and x < y. Over finite and over infinite This article is a revised and extended version of [Jurdzi´nski and Lazi´c 2007]. The second author was supported by a grant from the Intel Corporation. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee. c⃝20YY ACM 1529-3785/20YY/0700-0001 $5.00 ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY, Pages 1–23. 2 · M. Jurdzi´nski and R. Lazi´c data words, satisfiability for FO2(+1, <, ∼) was shown decidable and at least as hard as nonemptiness of vector addition automata. Whether the latter problem is elementary has been open for many years. Extending the logic by one more variable causes undecidability. Over data trees, FO2(+1, <, ∼) denotes a similar first-order logic with two vari- ables. The variables range over tree nodes, +1 stands for two predicates “child” and “next sibling”, and < stands for two predicates “descendant” and “younger sibling”. Complexity of satisfiability over finite data trees was studied by Boja´nczyk et al. [2009]. For FO2(+1, ∼), it was shown to be in 3NExpTime, but for FO2(+1, <, ∼), to be at least as hard as nonemptiness of vector addition tree automata. Decid- ability of the latter is an open question, and it is equivalent to decidability of multiplicative exponential linear logic [deGroote et al. 2004]. However, Bj¨orklund and Boja´nczyk [2007] showed that FO2(+1, <, ∼) over finite data trees of bounded depth is decidable. XPath [Clark and DeRose 1999] is a prominent query language for XML docu- ments [Bray et al. 1998]. The most basic static analysis problem for XPath, with a variety of applications, is satisfiability in the presence of DTDs. In the two ex- tensive articles on its complexity [Benedikt et al. 2008; Geerts and Fan 2005], the only decidability result that allows negation and data (i.e., equality comparisons between attribute values) does not allow axes which are recursive (such as “self or descendant”) or between siblings. By representing XML documents as data trees and translating from XPath to FO2(+1, ∼), Boja´nczyk et al. [2009] obtained a decidable fragment with

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